In the Griffiths and Harris, Principles of Algebraic Geometry, p.181, he states Kodaira Embedding theorem as follows :
Kodaira embedding theorem (Ver.1). Let $M$ be a compact complex manifold and $L \to M$ a positive line bundle. Then there exits $k_0$ such that for $k \ge k_0$, the map $\iota_{L^k} : M \to \mathbb{P}^N$ is well-defined and is an embedding of $M$ ; i.e., $L$ is ample (?).
On the other hand, in the Huybrechts's Complex geometry book he states Kodaira embedding theorem as follows :
Kodaira embedding theorem (Ver.2). Let $X$ be a compact Kahler manifold. A line bundle $L$ on $X$ is positive if and only if $L$ is ample. In this case, the manifold $X$ is projective.
Where the Kahler condition is used? I found that in the Huybrechts's proof of the theorem, he used the Kodaira vanishing theorem ( Proposition 5.2.2.) which is stated as follows :
Proposition 5.2.2 ( Kodaira Vanishing ) Let $L$ be a positive line bundle on a compact Kahler manifold $X$. Then $ H^q(X, \Omega_X^{p} \otimes L)=0$ for $p+q >n$.
Note that here the 'Kahler' condition is attached. But in the Griffiths's Proof (p. 190), he used other version of Kodaira-Nakano Vanishing theorem ( p.154) which is stated as follows :
Kodaira-Nakano Vanishing Theorem. If $L\to M$ is a positive line bundle, then $H^q(M, \Omega^p(L))=0$ for $p+q >n$.
A priori, it seems that in this theorem the condition Kahler is not attached but in Griffith's book p.148 he assumes that $M$ is a compact Kahler manifold. So there is a still possibiltiy of usage of the condition Kahler in the proof of Kodaira embedding theorem.
C.f. I am studying with next pdf note : https://www.math.stonybrook.edu/~cschnell/pdf/notes/complex-manifolds.pdf In p.95 of the note, Corollary 31.7, he states Kodaira vanishing theorem same as the Proposition 5.2.2. above without imposing the condition 'Kahler'.
Q. It's really true that the condition 'Kahler' is redundant in the above Kodaira Vanishing theorem (Proposition 5.2.2)? If 'Kahler' condition is deleted, then the Proposition 5.2.2 is false ?
And at p.96, theorem 32.3 in the note, he states Kodaira embedding theorem similar as the Griffith's Kodaira theorem (Ver.1.) without imposing the condition 'Kahler'.
And then, in his note p. 103, first paragraph after Class 35, he mentioned that "What we have shown is : a line bundle $L$ on a compact Kahler manifold $M$ is positive if and only if $L$ is ample", same as the Kodaira embedding theorem (Ver.2) above.
Where the Kahler condition is used? What is problem? Is there a point that I missed?
EDIT : Through discussion with Arctic Char, I think that reason of addition of the adjective 'Kahler' in the statement of Kodaira embedding theroem of Huybrechts's version is, to emphasize that $X$ is Kahler with respect to the Kahler metric whose existence is guaranteed by the existence of positive line bundle.
And..I don't understand why such existence of Kahler metric from existence of positive line bundle is true ; here by positivity of line bundle, we adopt definition of Huybrechts's : A line bundle $L$ is called positive if its first Chern class $c_1(L) \in H^2(X, \mathbb{R})$ can be represented by a closed positive real $(1,1)$-form. Here the positivity of a real $(1,1)$-form is defined as follows ( Huybrechts's book p.188, Definition 4.3.14 ).
Definition 4.3.14. A real $(1,1)$-form $\alpha$ is called (semi-)positive if for all holomorphic tangent vectors $0 \neq v \in T^{1,0}X$ one has
$$ -i \alpha(v,\bar{v}) >0 (\operatorname{resp.} \ge )$$
Can anyone helps?
Best Answer
This is quoting from the start of Class 31 in the note:
$\cdots$ (skipping several lines)
So, the mere existence of a positive line bundle on $M$ already implies the existence of a Kahler metric.